Does "transition model" alone in an MDP imply it's non-deterministic?
Yes, in which case, it will be more like a search problem, if it is not POMDP and have finite number of states. Or you can use the same framework (used for POMDP) with constrained (deterministic) transition matrix modeling for model-based systems.
If you think about it after you train any model/agent with an MDP modeling, during test time the optimal strategy is generally deterministic, i.e., given a feature/state you will take a particular action even if it has more than one non-zero element in each row of the transition matrix.